On free $${\mathbb{Z}_{p}}$$ actions on products of spheres

In this paper we consider free actions of large prime order cyclic groups on the product of any number of spheres of the same odd dimension and on products of two spheres of differing odd dimensions. We require only that the action be free on the product as a whole and not each sphere separately. In particular we determine equivariant homotopy type, and for both linear actions and for even numbers of spheres the simple homotopy type and simple structure sets. The results are compared to the analysis and classification done for lens spaces. Similar to lens spaces, the first k-invariant generally determines the homotopy type of many of the quotient spaces, however, the Reidemeister torsion frequently vanishes and many of the homotopy equivalent spaces are also simple homotopy equivalent. Unlike lens spaces, which are determined by their ρ-invariant and Reidemeister torsion, the ρ-invariant here vanishes for even numbers of spheres and linear actions and the Pontrjagin classes become p-localized homeomorphism invariants for a given dimension. The cohomology classes, Pontrjagin classes, and sets of normal invariants are computed in the process.